If the oil light wasn't coming on before the oil change, make sure you used the correct oil type. Using an oil different of a different viscosity than the manufacturer recommended oil can cause the light to come on. If the oil gauge is not reading at all, then you may have an electrical problem and the oil pressure may be OK. Make sure the wire to the oil pressure switch didn't become disconnected during the oil change.
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If the blower motor is drawing too much current, it could be ruining your blower motor resistor. You can check the amperage draw on the motor, but you would need a spec to know if it is over the limit, and that may be difficult to find.
Read our article on the check engine light in our Encyclopedia to understand why the light comes on. There are many possibilities. You will need to have the codes checked to know what the possibilities are. If you can get the codes, some places parts stores may do this for free, post them here and someone may have an answer for you.
Try checking the power and ground terminals at the OBD connector. Connect a voltmeter between terminals #5 and #16 and you should have battery voltage. Search "GM OBDII connector pinout" and you will find pictures of the terminal locations.
copy and paste this link for a picture of the vehicle speed sensor for your car - http://images.channeladvisor.com/Sell/SSProfiles/60000150/Images/3/MSC901.jpg
Is the radio turning on? Is there anything from the display on the radio or is it blank?
This may be a difficult problem to locate. It sounds like there is a poor electrical connection in the circuit for the speedometer. There is a speed sensor on the transmission, the speedometer and the wiring in between them and the problem can be with any of these.
If your cooling fan motor keeps running, you will want to disconnect remove the fuse (40 amp fuse in the battery junction box) to keep it from draining the battery. It's likely that the CCRM (Constant Control Relay Module) is bad and keeping the fan on.
At this point you should have the codes checked to determine why the check engine light is on.